Here we consider an atomic system weakly interact with the external quantum fields. These
systems are frequently known as open-quantum system [1]}, [2]}.
The reduced dynamics of the system sharply depend on the field correlation
function. There exist several parallel approaches to calculate the atomic
energy level shift due to vacuum fluctuations [3]}.
Dalibard-Dupont-Cohen (DDC) calculated the rate of change of the mean atomic
energy, which basically determines spontaneous excitation rates and radiative
reactions under the vacuum fluctuations [4]}. Similar methods
based on Langevin dynamics are used to calculate the fluctuation induced
interactions [5]}. The quantum master equation (QME) is also an important
tool to derive the reduced dynamical equation of the atoms
[6]}, [7]}. Benatti first
described the positive time-evolution of the quantum systems weakly coupled
with a scalar field in the Minkowski vacuum [8]}. Later
multiple attempts by using the QME formalism are found to analyze the Casimir
physics from scratch [9]}, [10]}, [11]}.
The system-field weak coupling Hamiltonian gives an effective second-order
contribution in the QME. It consists of two parts, the real part gives a
dissipative dynamics, and the imaginary terms give a second-order shift
term of atomic levels. This shift is known as Lamb shift [1]}. If the
atoms are interacting with the same external field, then there is a possibility
of the creation of field-induced entanglement, which is known as the
common-environment effect [13]}, [14]}. The
individual system-field interaction cross-terms in the second order of QME also
produce a shift term in the dynamics which are particularly dependent on the
distance between the two atoms. As a result, there is a formation of
inter-atomic correlation in the dynamics. This interaction is called resonance
Casimir Polder interaction (RCPI) [9]}. One of the atoms are
kept in a ground state and the other in an excited state. Due to the
interaction with the vacuum of the same quantized field, the exchange of real
photons occurs, and we are getting a non-zero expectation value of the
correlated state. This phenomena gives rise to resonance interaction
[16]}.
